After Ashley Young atoned for missing a penalty by wiping out Mark Noble's first-half spot-kick, both sides looked like they would have to settle for their third successive draw, but the 20-year-old substitute had other ideas as he struck a 93rd-minute winner.
Zola had seen his team throw away a two-goal lead at the Stadium of Light on Saturday and the Italian responded by making three changes from the side that finally drew 2-2 with Sunderland, as Julien Faubert, Manuel Da Costa and Scott Parker replaced Jonathan Spector, James Tomkins and the suspended Radoslav Kovac.
Without a Premier League victory since their win at Wolverhampton Wanderers on the opening day of the campaign, West Ham kicked off looking for their first home success of the season.
And after a cagey opening quarter-hour during which Spector replaced the injured Herita Ilunga, Noble finally teased the first serious save of the evening out of Brad Friedel, when he cut into the area and unleashed an awkward 15-yarder, that the Villa keeper parried to safety.
On 20 minutes, Robert Green was also called into action, when Young's corner was cleared to Stiliyan Petrov, whose stinging 25-yarder deflected off Da Costa before being expertly tipped over at the last second by the alert England goalkeeper.
Martin O'Neill had made just two switches from side that had drawn 1-1 at Everton on Saturday as Habib Beye and John Carew replaced James Milner and the banned Carlos Cuellar.
Shortly afterwards, Gabriel Agbonlahor unleashed an angled scorcher that Green beat aside for a Villa corner and, as the visitors started to gain the upper hand, there was more bad news for Zola, when top-scorer Carlton Cole hobbled down the tunnel under the worried gaze of the onlooking Fabio Capello.
Five minutes before the break, Green again came to the Hammers' rescue, when he held Carew's clever, close-range, shot-on-the-turn but just as the opening period looked like ending goalless, West Ham broke the deadlock.
With referee Steve Bennett counting down the seconds, Cole's replacement, Hines, hit the damp East End turf, when he tangled with the consequently booked Beye and Noble claimed his second goal of the season with an unstoppable penalty that gave the wrong-footed Friedel absolutely no chance.
The Villa remonstrations were still going on as the teams headed off at the break but after introducing former Hammers' skipper, Nigel Reo-Coker for the restart, in place of Emile Heskey, the Midlanders were given a great opportunity to level, when they were awarded an equally dubious spot-kick.
This time it was West Ham's turn to lodge the vehement protests after the soaring Da Costa was harshly adjudged to have fouled ex-Hammer James Collins but Green guessed correctly, as he dived at the base of his left-hand post to block Young's 49th-minute penalty.
The Villa wide boy's agony was then compounded by a Parker lunge that earned the Hammers' combative midfielder an incredible, eighth yellow card of the season.
Young soon silenced the home cheers, though, when he cut in from the left touchline and threaded a beautiful 20-yarder over the groping Green and just under the keeper's left-hand angle to atone for that penalty miss.
In reply the lively Hines fired a couple of efforts narrowly across goal and then the young Hammers' substitute forced the flustered Friedel to claw out his angled shot as West Ham desperately tried to regain their advantage.
Beye's late foul on Spector then earned the Senegal international a second yellow card of the night and, as Villa's ten men held on for a draw, Hines ensured that West Ham claimed that elusive first home win of the season, when he clipped an angled shot over Friedel with just seconds remaining.